The present invention relates generally to three-dimensional (3D) computerized tomography (CT) and, more particularly, to methods and apparatus for converting x-ray cone beam data to planar integrals for 3D image reconstruction through inverse Radon transformation.
In conventional computerized tomography for both medical and industrial applications, an x-ray fan beam and a linear array detector are employed. Two-dimensional (2D) imaging is achieved. While the data set is complete and image quality is correspondingly high, only a single slice of an object is imaged at a time. When a 3D image is required, a "stack of slices" approach is employed. Acquiring a 3D data set a 2D slice at a time is inherently tedious and time-consuming. Moreover, in medical applications, motion artifacts occur because adjacent slices are not imaged simultaneously. Also, dose utilization is less than optimal, because the distance between slices is typically less than the x-ay collimator aperture, resulting in double exposure to many parts of the body.
A more recent approach, based on what is called cone beam geometry, employs a two-dimensional array detector instead of a linear array detector, and a cone beam x-ray source instead of a fan beam x-ray source. At any instant the entire object is irradiated by a cone beam x-ray source, and therefore cone beam scanning is much faster than slice-by-slice scanning using a fan beam or a parallel beam. Also, since each "point" in the object is viewed by the x-rays in 3D rather than in 2D, much higher contrast can be achieved than is possible with conventional 2D x-ray CT. To acquire cone beam projection data, an object is scanned, preferably over a 360.degree. angular range, either by moving the x-ray source in an appropriate scanning trajectory, for example, a circular trajectory around the object, while keeping the 2D array detector fixed with reference to the source, or by rotating the object while the source and detector remain stationary. In either case, it is relative movement between the source and object which effects scanning.
Most image reconstruction procedures in x-ray CT are based on the Radon inversion process, in which the image of an object is reconstructed from the totality of the Radon transform of the object. The Radon transform of a 2D object consists of integrals of the object density on lines intersecting the object. The Radon transform of a 3D object consists of planar integrals. The cone beam data, however, are not directly compatible with image reconstruction through inverse Radon transformation, which requires the sue of planar integrals of the object as input. Consequently, image reconstruction by inversion from cone beam scanning data generally comprises two steps: (1) convert the cone beam data to planar integrals, and (2) perform an inverse Radon transform on the planar integrals to obtain the image. The present invention is primarily directed to efficient methods and apparatus for converting x-ray cone beam data to planar integrals, or values representing planar integrals, on a set of arbitrary planes in Radon space. The above-incorporated application Ser. No. 631,818 [RD-19564] discloses a two-step method for performing an inverse Radon transform starting with planar integrals on a set of coaxial vertical planes in Radon space. Thus the invention disclosed herein may be employed to convert x-ray cone beam data to values representing planar integrals on a set of coaxial vertical planes in Radon space, and the invention of application Ser. No. 631,818 [RD-19564] may be employed to perform the inverse Radon transformation portion of the 3D image reconstruction.
One method for converting cone beam data to planar integrals is disclosed in Gerald N. Minerbo, "Convolutional Reconstruction from Cone-Beam Projection Data", IEEE Trans. Nucl. Sci., Vol. NS-26, No. 2, pp. 2682-2684 (Apr. 1979). Unfortunately, as is discussed, for example, in L. A. Feldkamp, L. C. Davis, and J. W. Kress, "Practical Cone-Beam Algorithm", J. Opt. Soc. Am. A., Vol. 1, No. 6, pp. 612-619 (Jun. 1984), the derivation in Minerbo contains an error which cannot easily be rectified and which renders the result invalid.
In Bruce D. Smith, "Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods", IEEE Trans. Med. Image., Vol MI-44, pp. 1425 (Mar. 1985), there is disclosed a method for converting from cone beam data the one-dimensional convolution of the planar integrals with the Horn's kernel. Since the convolution mixes together the planar integrals on all the planes, the computation of one point of the convolved result requires all the data on the detector at one view angle. Thus the task is very computationally intensive.
In P. Grangeat, "Analysis of A 3D Imaging System by Reconstruction from X Radiographies in Conical Geometry" ("Analyse d'un System D-Imagerie 3D par Reconstruction a partir de Radiographies X en Geometrie conique"), Ph.D. Thesis, National College of Telecommunications (I-Ecole Nationale Superieure des Telecommunications), France (1987), a technique is disclosed for computing the derivative of the planar integrals from cone beam data. The computed data points, however, reside on a set of great circles on a spherical shell in Radon space. These great circles in general do not fall on any arbitrary set of planes in Radon spaces, and do not fall on a set of coaxial vertical planes in Radon space. Thus they are not suitable for input to inverse Radon transformation. It would require an extensive effort in three-dimensional interpolation to get the data on the vertical planes to be sued in inverse Radon transformation, and furthermore interpolation would introduce errors into the data.